'AC/DC Circuits' printed from http://nrich.maths.org/

Ohm's law forms the basis for all DC electrical circuit analysis.
It was first suggested by Georg Simon Ohm in 1826 that the current
flowing in a uniform conducting wire is directly proportional to
the potential difference between its ends. The constant of
proportionality relating V and I is known as the resistance R. The
unit of resistance is the ohm ($\Omega$).

$V = I R$,

where

$R = \frac{\rho}{LA}$,

$\rho$ is the resistivity of the material,
$A$ is the cross-sectional area of the component,
$L$ is the length of the component.

In the simple circuit shown, a battery (or source) delivers current
to a resistor. Arrows are used to indicate the sense of the voltage
and the direction of the current. Although electrical current is
actually the flow of negatively charged electrons, by convention,
current is said to flow from high to low potential, as though it
were composed of positive charges.

Non Ohmic Devices

Many electrical devices have I-V characteristics that vary in a non
linear fashion. Such devices are termed non-linear and do not obey
Ohm's law. Examples include filament lamps, diodes and thermistors.
The resistance of a non ohmic device may vary with time.

Electrical resistance is a consequence of electrons colliding with
ions and losing mechanical energy.

For the case of the filament bulb, an increasing voltage causes
the temperature of the filament to rise and as a result the ions
begin to vibrate with greater amplitude. This leads to a higher
frequency of collisions between electrons and ions; the resistance
of the component has hence increased.

A thermistor is a resistor whose resistance varies with
temperature. There are two possibilities:
Negative Temperature Coefficient Thermistor: A resistor whose
resistance decays with increasing temperature.

NTC thermistors are typically constructed from semi-conductor
materials. As voltage increases the temperature of the
semi-conductor rises, energy becomes available to liberate
electrons from their atoms, the charge carrier density and hence
the current increases in a non-linear fashion with voltage.

Electrical Power

$P = IV = I^2 R = \frac{V^2}{R}$

Electrical power is the rate at which electrical energy is
transferred. Its units are joules per second, or watts.

Resistors connected in
series:

Resistors connected in series will have a common current passing
through them. The voltage drop across each resistor may be
different but all such voltages will sum to the voltage of the
supply.

$V_{total} = V_1 + V_2 + ... +V_n$

$I R_{total} = I R_1 + I R_2 + ... +I R_n$

$R_{total} = R_1 + R_2 + ... +R_n$

Resistors connected in
parallel:

Resistors connected in parallel will have a common voltage across
them but may have a different current through them.

Kirchoff's voltage law is a statement of energy conservation; it
states that around any closed loop the sum of the EMFs is equal to
the sum of the voltage drops. If we define voltage drops as
negative and EMFs as positive, then around any closed loop the sum
of the voltages equals zero.

Kirchhoff's Current
Law:

Kirchhoff's current law states that the sum of the currents into a
node equals the sum of the currents leaving the node. This is a
consequence of the fact that charges do not accumulate at nodes. If
we define inflows as positive and outflows as negative, then we can
say that the sum of the currents into a node is equal to
zero.

Mesh/Loop current
Analysis

Mesh/Loop current analysis are the application of Kirchhoff's
voltage law to solve for unknown currents. In mesh current analysis
we assign unknown currents to each element in the circuit. We then
apply current conservation at each node. In Loop current analysis
we assign unknown currents to each loop, the total current passing
through a component is therefore the vector sum of the loop
currents through this element. The number of independent equations
and unknown will be equal to the number of independent loops.

Question:
Using Kirchhoff's laws, find the current flowing through each of
the components in the diagram below.

Solution:

$\sum_{node\ a} I = I_1 - I_2 - I_3 = 0$

$\sum_{loop\ 1} voltages =V_1 - I_1R_1- I_2R_2 = 0 $

$\sum_{loop\ 2} voltages = - I_2R_2- I_3(R_3 + R_4) = 0 $

We therefore have 3 independent equations and 3 unknowns.

Using equation 1 we can eliminate $I_1$ from equation 2

$I_1 = I_2 + I_3$

Equation 2 then becomes: $V_1 - I_3R_1 - I_2(R_1+R_2) = 0$

If we now solve this equation simultaneously with equation 3 we
find that:

$I_1 = \frac{3V_1}{5R}$

$I_2 = \frac{V_1}{5R} $

$I_3 = \frac{2V_1}{5R}$

AC
Circuits

Most electric and magnetic devices use alternating or fluctuating
currents. The two main reasons for this are concerned with power
transmission and information processing. Firstly, an alternating
electric power may be made available at almost any desired voltage
and current level by the use of a transformer. Secondly,
fluctuating currents serve a great purpose in information
transmission. For example, a microphone will convert the spoken
word into a current which has frequency equal to that of the
speech. We are hence able to convert physical information into
electrical data which we may then process. It is often useful to
consider an alternating voltage or current as a sinusoid. In fact,
this may not always be the case; the current through a microphone
will often take some other form. However, it is true that we may
represent any periodic function as a infinite series of sinusoids.
This is known as a Fourier series. The Fourier series will not be
discussed further here but it may be of further interest to the
reader.

We often represent voltage in the form:

$V = V_0 \cos(2\pi f t)$,

where $V_0$ is the peak voltage and $f$ is the frequency of
oscillation.

Such a voltage would cause a current of the form:

$I = I_0 \cos(2\pi ft + \phi)$,

where $I_0=\frac{V_0}{R}$ is the peak current and $\phi$ is the
phase of the current with respect to the voltage.

If the load is completely resistive (there is no inductive or
capacitive component) then the voltage and current will be in phase
and $\phi = 0$.

We know that the impedance of a resistor is simply R, this is
purely resistive; in complex form $Z = R + 0i$.

Capacitors:

The impedance of a capacitor is defined: $X_c = \frac{1}{2\pi i f
C}$, where $f$ is the frequency of the alternating current, $C$ is
the magnitude of the capacitance and $i =\sqrt{-1}$

We see that the impedance is frequency dependant; the impedance of
a capacitor is inversely proportional to the frequency. This
frequency dependence provides capacitors with a filtering ability;
they effectively filter low frequency signals whilst allowing high
frequency to pass. The two fundamental uses of a capacitor include
coupling and bypass capacitors.

Coupling Capacitor:
Coupling capacitors are placed in series with a component, its
purpose is to filter DC signals whilst allowing AC signals to
pass.

Bypass Capacitor:
Bypass capacitors are connected in parallel with a component, at
high frequencies they short circuit this component (the current is
allowed to bypass). The bypass capacitor conducts an alternating
current around a component whilst allowing DC through it, it can be
used to filters electric noise caused by ripple voltages.

Inductors:

The impedance of an inductor is defined as $Z = 2\pi i f L$

The impedance of an inductor behaves in the opposite manner to the
capacitor - its impedance is proportional to frequency.

Electrical Filters

The frequency dependent characteristics of inductors and capacitors
enable the creation of many useful circuits, these include high
pass, low pass and band pass filters.

Low Pass filter:
A low pass filter is a circuit which allows easy passage of low
frequency signals but prevents the passage of high frequency
signals. A simple low pass filter is shown below.

High Pass filter:
A high pass filter is a circuit which allows easy passage of high
frequency signals but prevents the passage of low frequency
signals. The circuit is shown below.

Band Pass filter:
By combining the properties of the high pass and the low pass
filter we can accomplish a band pass filter. At low frequency the
majority of the input will be dropped across the capacitor whilst
at high frequency it will be dropped across the inductor. There
exists a small range of frequencies at which an appreciable voltage
will be output, this occurs when the impedance of the capacitor
matches the inductor and is known as the resonant frequency.